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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(28665, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,21,2,35]))
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pari:[g,chi] = znchar(Mod(7849,28665))
\(\chi_{28665}(1369,\cdot)\)
\(\chi_{28665}(5464,\cdot)\)
\(\chi_{28665}(7849,\cdot)\)
\(\chi_{28665}(9559,\cdot)\)
\(\chi_{28665}(11944,\cdot)\)
\(\chi_{28665}(13654,\cdot)\)
\(\chi_{28665}(16039,\cdot)\)
\(\chi_{28665}(17749,\cdot)\)
\(\chi_{28665}(20134,\cdot)\)
\(\chi_{28665}(21844,\cdot)\)
\(\chi_{28665}(24229,\cdot)\)
\(\chi_{28665}(28324,\cdot)\)
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sage:chi.galois_orbit()
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pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\((25481,11467,18721,11026)\) → \((1,-1,e\left(\frac{1}{21}\right),e\left(\frac{5}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) | \(29\) |
| \( \chi_{ 28665 }(7849, a) \) |
\(1\) | \(1\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{4}{21}\right)\) |
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sage:chi.jacobi_sum(n)
